One of the reasons why the Fourier transformation is a useful tool is that it converts differentiation (hard to understand) to multiplication (easier to understand). Many of the basic proofs of Fourier theory use complex analysis.
This is a beautiful theory, but unstable under perturbations (for instance, to work on manifolds), as it relies on exact computations that cannot be carried out when things change a little.
In recent years, there has been interest in avoiding the use of complex analysis.
The talk explains some of the progress and the difficulties.
Pure Maths Seminar
Tue, 20/03/2012 - 12:00pm to 1:00pm
RC-4082, Red Centre, UNSW